Enhanced Flexibility of a Bulged DNA Fragment from Fluorescence

Jan 17, 1998 - By employing Brownian dynamics simulations, it is shown that the FPA results from the fragment with the double insertion can be interpr...
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Macromolecules 1998, 31, 695-702

695

Enhanced Flexibility of a Bulged DNA Fragment from Fluorescence Anisotropy and Brownian Dynamics Maddalena Collini,† Giuseppe Chirico,† Giancarlo Baldini,*,† and Marco E. Bianchi‡ Dipartimento di Fisica, Istituto Nazionale di Fisica della Materia, Universita` degli Studi di Milano, via Celoria 16, Milano 20133, Italy, and DIBIT, San Raffaele Scientific Institute, Milano 20132, Italy Received July 28, 1997; Revised Manuscript Received November 18, 1997

ABSTRACT: A fragment of DNA built from 50 base pairs and two five-adenine unpaired sequences, each inserted in the center of the duplex, has been investigated by measuring the polarization anisotropy decay of intercalated ethidium fluorescence (FPA). When the data from this bulged fragment are compared to those from a straight fully paired duplex or to those from a fragment bent by a five-adenine insertion in one strand, then systematic differences are found in the fluorescence parameters. By employing Brownian dynamics simulations, it is shown that the FPA results from the fragment with the double insertion can be interpreted by assuming a softening of the elastic constant at the bulge. Estimates of the torsion and the bending rigidities at the bulge site are also given.

Introduction Fluorescence polarization anisotropy (FPA) has proved to be a sensitive technique to investigate the size and the conformation of short DNA fragments labeled with suitable fluorescent probes.1-3 When short fragments are studied, the most relevant sources of fluorescence depolarization are the rigid body global rotations (spinning and tumbling diffusion) whereas a smaller contribution is given by internal motions dynamics (torsions and bending). In order to better understand the weight of the different contributions, it has been found enlightening to develop proper models of the DNA dynamics to be probed by FPA, since by means of this approach one can extract quantitative information from the experimental data. When regular DNA fragments are considered, suitable and robust analytical theories are available.4-6 On the other hand, Brownian dynamics (BD) simulations are very useful to analyze FPA data since no accurate analytical model appears to be available for “anomalous” structures. By modeling the DNA as a chain of touching beads, one can show that the FPA correlation function can be successfully predicted from the comparison of simulations and data. This paper follows the investigation on short DNAs with structural anomalies, which was previously applied3 to the case of a 50-base pair duplex containing the insertion of an extra five-adenine sequence in one strand and the consequent bending of the structure was established by FPA measurements and BD simulations. Here we apply this analysis to a 50-base pair fragment that contains two sequences of five adenines each inserted in the middle of the duplex and expecting a bulge to occur. While a single insertion has been shown to lead to a stable curvature as the main effect, it is reasonable that the bulge, induced by the double insertion, should enhance the flexibility in the unpaired region.7 The effect of a 3-base pair bulge on circularization kinetics of small DNA circles has been shown to lead to estimates of the torsional and bending rigidities at the bulge site, which were found to be much lower † ‡

Universita` degli Studi di Milano. DIBIT.

than those typical of the duplex.7 The effect of loops and bending on short DNA regions has been examined in recent years with a variety of techniques, such as denaturation kinetics,8 gel migration,9 circularization kinetics,7 fluorescence resonance energy transfer,10 and fluorescence polarization anisotropy.3 Bendings and loops are suggested to be correlated to base unpairing and to act by favoring the binding of proteins or the contact between otherwise distant regions of the DNA helix.11,12 The aim of the paper is to ascertain whether and to what extent the FPA technique is sensitive to the presence of both dynamic and structural anomalies in short DNA fragments and, by combining the experimental data to BD simulations, to give a quantitative estimate of the induced changes in the DNA elastic properties and the equilibrium structure. The paper first gives a summary of FPA and BD theories. Then, in the Experimental Section, details on the preparation of DNA samples and their sequences are reported. The procedure for FPA measurements and the parameters for BD simulations are shown in the next sections. In the Results and Discussion section, after presentation of the characterization of the two fragments by circular dichroism and UV absorbance melting curves, the FPA data, fitted by a simple mechanical model and by BD simulations, are discussed in detail. The separate contribution of torsional and bending motions to FPA is estimated by means of simulations under different conditions. Theory A. Fluorescence Polarization Anisotropy. Fluorescence polarization anisotropy decay of ethidiumDNA complexes has been widely employed in order to gain information on the internal dynamics of nucleic acids.2,3,13 For this reason, we recall the main concepts of the FPA theory and show here only the most relevant equations. The anisotropy of fluorescence, r(t), can be derived by detecting its time decay at the two standard polarizations (parallel and perpendicular to the vertical polarization of the exciting beam, I|(t) and I⊥(t), respectively). With the exception of very short times when

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wobbling may occur (less than 1 ns), the probe can be considered firmly bound to the macromolecule and its binding site geometry well-known. As a consequence, the fluorescence depolarization at different times contains information about the macromolecular motion. In order to gain quantitative information on the DNA dynamics, one should employ a proper model of the polymer diffusive motions. Some detailed analytical models are available. In this case, a mean cylindrical symmetry for DNA is assumed which is described as a deformable elastic filament6 with a straight equilibrium structure. A general assumption regarding the uncoupling of intercalating dye wobbling, torsional, and bending motions is made which leads to the expression of the FPA:6

r(t) )

(I|(t) - I⊥(t)) (I|(t) + 2I⊥(t))

2

) r0

∑ In(t)Cn(t)Fn(t)

(1)

n)-2

This model allows the factorization of the different contributions to the molecular motion: In(t) are the internal correlation functions that depend only on the angle between the dye transition dipole and the polymer axis of symmetry ()70.5°) and, since the fluorophore wobbling dynamics spans a range (ps) to which our FPA setup is not sensitive, are essentially time independent. The depolarization due to the fast wobbling is described by a reduced limiting anisotropy factor r0 = 0.36. Cn(t) represent the twisting correlation functions, related to torsional motions, whose longest motion is the diffusion spinning around the helix axis: Cn(t) ) exp[-n2〈∆φT(t)2〉], where ∆φT(t) is the torsional displacement of the subunits around their z-axes (here identified as the cylinder’s long symmetry axis). Fn(t) are the tumbling correlation functions, related to bending motions, whose longest motion is the (doubly degenerate) tumbling motion around an axis perpendicular to the helix axis x: Fn(t) ) exp[-(6 - n2)〈∆φB(t)2〉], where ∆φB(t) is the bending angular displacement. Explicit expressions for the correlations functions can be found in ref 2. In the presence of a small free dye contribution, the total anisotropy is given by the intensity-weighted average of the bound and of the free fluorophore.14 The free dye anisotropy is given by rF(t) ) r0e-t/τrot, where τrot is taken15 to be 100 ps and r0 ) 0.4. The FPA data shown in the results section are obtained by detecting the FPA decay in the frequency domain, an alternative method equivalent to the time response.16,17 It may help, however, to recall some relevant quantities that are directly obtained from the measurements and to show how they compare with eq 1. In the frequency domain, a radio frequency (MHz) modulated beam impinging on a sample forces the fluorescence emission to be modulated at the same frequency, although phase shifted with respect to the excitation and reduced in its relative amplitude (demodulation). Phase shift and demodulation are recorded at each frequency for each polarization direction. It is common to represent the data by collecting the phase shift differences and the demodulation ratios between the two polarizations, namely,

∆φ ) φ⊥ - φ|

∆M ) M⊥/M|

tg(∆(ω)) ) Im(L(I⊥)(ω))Re(L(I|)(ω)) - Re(L(I⊥)(ω))Im(L(I|)(ω)) Re(L(I⊥)(ω))Re(L(I|)(ω)) + Im(L(I⊥)(ω))Im(L(I|)(ω)) ∆M(ω) )

|L(I|)(ω)|

(3)

where Re(L(I⊥,|)(ω)) and Im(L(I⊥,|)(ω)) represent the real and the imaginary parts of the Laplace transforms. We recall that I|(t) ) Itot(1 + 2r(t))/3 and I⊥(t) ) Itot(1 - r(t))/ 3. In this way, the fluorescence anisotropy decay in the time domain (eq 1) can be related, through eq 3, to phase differences and demodulation ratios in the frequency domain.15 B. Brownian Dynamics. Brownian dynamics simulations have been performed with a FORTRAN program that integrates the Langevin equations for the polymer dynamics with a second-order approximation.18 The general procedure is based on some basic equations by Ermack and McCammon and Fixman.19,20 In this paper we adopt the algorithm and the formalism applied by Allison21 to DNA and further extended by Chirico and Langowski;22,23 only the essential background is given here. DNA segments are modeled as a string of touching beads, and the bending and torsional kinematics are described by assigning the positions of the center of each of the M beads and the direction of M - 1 reference directions perpendicular to the bonds. The hydrodynamic interactions are described by the the RotnePraeger tensor.24 The coupling of the torsional and bending dynamics is applied as in ref 22. The diffusive molecular properties are introduced in the simulations via the bead’s translational diffusion coefficient D0 and the rotational spinning diffusion coefficient of each bond DR. The choice of the diffusion coefficient is made by keeping the total volume of the string of beads equal to that of the DNA cylinder. This equivalence has been proposed by Hagerman25 and gives a satisfying agreement between the tumbling diffusion coefficient of the bead chain and of the corresponding cylinder.26 As an example, a DNA radius R ) 1.3 nm would correspond to a string of beads chain with bead’s radius RBEADS ) 1.592 nm. This scaling imposes some coarseness to the model since only DNA lengths proportional to the bead’s radius are possible. The spinning diffusion coefficient is more delicate since, instead of the beads, the spinning unit is the bead-to-bead bond. We assume a constraint on the total spinning diffusion of the DNA cylinder. From the torsional Langevin dynamics one would predict a total spinning diffusion for the fragment:

Dspin ) DR/(M - 1)

(4)

where DR is the spinning diffusion coefficient of the single bond. This assumes negligible effect of the bending-torsional coupling on the rigid body spinning dynamics.22 On the other hand, one knows accurate expressions of Dspin such as those given by Garcia de la Torre,27 where Dspin depends on the total DNA length (L) and radius (R):

(2)

these quantities can be related to the Laplace transform of the intensity decays at the two polarization conditions, L(I⊥,|)(ω) in the following way:15

|L(I⊥)(ω)|

Dspin )

KBT [3.841πηLR2(1+δ|)] δ| ) (0.677/p) - (0.183/p2) (5)

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Flexibility of a Bulged DNA Fragment 697

where p ) L/2R is the axial ratio and η is the solution viscosity at temperature T. The DR value is found by inversion of the equation obtained by comparing the BD (eq 4) and the hydrodynamic (eq 5) values of the spinning coefficient. As an example, for a 50-bp DNA simulated with M ) 5 beads of b0 ) 3.184 nm of diameter and DR ) 50.15 MHz, corresponding to the spinning of a rod 0.34 nm high and with a radius of =0.65 nm. Harmonic intermolecular potentials take into account the rigidities among the beads through a stretching potential Us, a bending potential Ub, and a torsional potential Ut. We give for completeness the form of these potentials:

Us )

KBT M-1 (bi - b0)2 2 i 2δ

∑

Ub )

Ut )

KBT M-2 1

∑ i)1

2

KBT M-2 1 2

∑ i)1

βi2

(7)

(Ri + γi)2

(8)

2

ψi

2

ξi

(6)

In the formulas, δ is the stretching variance and ψi are the bending variances, defined in terms of the polymer local persistence lengths Pi through ψi ) (b0/Pi)1/2 while the torsional variances, ξi, are defined through the local torsional rigidities Ci, ξi ) (b0KBT/Ci)1/2. In this paper, the torsional and bending parameters are position dependent and they are included in the sum over the number of beads in order to make it possible to employ, when needed, different values of and for each bead. The computation of the FPA correlation function throughout each simulation is accomplished by performing the following average:3

rB(jδt) )

1

1

(Nm - j - 1) (M - 1)

M-1 Nm-j

P2(cos(Φi,j,k)) ∑ ∑ i)1

k)1

(9)

where P2 indicates the second Legendre polynomials, Nm is the number of time steps, M is the number of j k(i + j)dt)µ j k(idt) with µ j k(t) beads, and cos(Φi,j,k) ) µ indicating the direction of the intercalated ethidium transition dipole at the k-th position. Experimental Section A. Materials. DNA oligonucleotides have been synthesized with the phosphotriester method, as previously described,3,28 and successively purified by gel electrophoresis. The control fragment F0 is obtained by the reaction of seq.0 with seq.0c, yielding a 50-base pair fragment.

seq.0:

CCTCGAGAAGCTTCCGGTAGCAGCCCCTTGCTAGGACGGATCCCTCGAGG

seq.0c:

GGAGCTCTTCGAAGGCCATCGTCGGGGAACGATCCTGCCTAGGGAGCTCC

The bulged fragment F2 is obtained with the same sequence seq.0 synthesized with five extra adenines in the middle (for a total of 55 bases). This sequence seq.5 reacted with a complementary sequence (the same as seq.0c) with the same five extra adenines in its center (seq.5c), as shown below.

seq.5: CCTCGAGAAGCTTCCGGTAGCAGCCAAAAACCTTGCTAGGACGGATCCCTCGAGG seq.5c: GGAGCTCTTCGAAGGCCATCGTCGGAAAAAGGAACGATCCTGCCTAGGGAGCTCC We recall that the kinked fragment, which we will refer to as F1 in the results and discussion and previously studied in our laboratory,3 was obtained by the reaction of seq.5 with seq.0c. The three different samples have 50 paired bases each; in addition, F1 has a single insertion of five unpaired bases in one strand, while F2 has a double insertion or “bulge” at the center of the double filament. The DNA fragments have been purified by gel electrophoresis for whose detection an aliquot of each preparation was 5′-labeled with [32P]ATP and T4 DNA kinase (Boehringer). Calf thymus DNA from Sigma Chemical Co. has been employed in control measurements when needed. Ethidium bromide (EB) was purchased by Sigma Chemical Co. as a powder and freshly dissolved in the buffer. It was added to DNA samples in the concentration of 1 EB/200 base pairs. Both DNA and EB were dissolved in STE100Mg buffer (10 mM Tris-HCl a pH 7.4, 100 mM NaCl, 5 mM MgCl2, 1 mM EDTA). EB and DNA concentrations were estimated by spectrophotometric measurements in cuvettes of 1-cm optical length and quartz walls. The values of the molar extinction coefficients used were  ) 6600 cm-1 M-1 for DNA at 258 nm and  ) 5860 cm-1 M-1 for EB at 478 nm. B. Methods. Melting profiles have been acquired with a Lambda 2 (Perkin Elmer) spectrophometer equipped with a thermostated cell holder. The temperature was controlled by a Haake F3 circulating bath and read by a thermocouple in the sample cell. Time-driven scanning of the absorbance was used to check the stabilization of the system at a given temperature; then the spectrum was recorded. Circular dichroism spectra were recorded with a Jasco J-500A spectropolarimeter directly interfaced to a personal computer. Each acquisition is the average of 40 data points at each wavelength with a spacing of 1 nm. Fluorescence polarization anisotropy decay measurements have been obtained with a frequency domain K2-ISS (Urbana, IL) fluorometer operated in the range from 1 to 40 MHz. Light excitation was accomplished by the green 514.5-nm output of an argon ion laser (Spectra Physics 2025) employing a power of 500-700 mW. For further details, see ref 15. Digital data acquisition and storage was provided by the ISS-A2D ACD card inserted in a personal computer. For each set of data, 25 logarithmically spaced frequencies are employed in the range 1-40 MHz with a cross-correlation frequency of 80 Hz. A 550-nm long-pass filter was used to cut the scattered light from excitation. The temperature was kept nearly constant by a Haake F3 thermostating bath and directly checked by a thermocouple inserted in the cell compartment which gave the value of T ) 24.5 °C. Data are analyzed according to a FORTRAN routine.3 The number of the fitting parameters is dependent on the fitting model; however, EB lifetimes are always kept at 22.5 ns and at 1.8 ns for the bound and the free species, respectively. Data fitting is accomplished by minimizing the total χ2, i.e., the sum of the square of the normalized differences between the experimental and computed phase differences and modulation ratios. The trial phase differences and modulation ratios are the Laplace transform of the fluorescence intensities obtained from either an analytical r(t) for the flexible joint model or a simulated r(t) for the BD simulations (see eq 1).3 Brownian dynamics simulations have been performed on a HP 9710 Apollo workstation and on Cray C92 (Cineca, Istituto Nazionale di Fisica della Materia) with FORTRAN 77 programs developed in our laboratory.22,23 BD simulations have been obtained by computing trajectories of 750.000 steps, each time step being of 6.7 ps, for a total of 5000 ns, in order to calculate the fluorescence polarization anisotropy decay correlation function which becomes negligible at times longer than 200 ns. For each simulation, the following values of the parameters have been kept fixed: solution temperature T )

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Figure 1. Molar ellipticity of the reference F0 fragment (open circles) and of the double-inserted F2 fragment (filled circles) versus wavelength. The dotted line represents the difference between the two spectra.

Macromolecules, Vol. 31, No. 3, 1998

Figure 2. Differential UV absorption at 258 nm versus temperature for the reference F0 (open circles) and the doubleinserted F2 fragment (filled circles).

297.15 K, solution viscosity 0.9111 cp, and stretching parameter 0.01, which has been found suitable22 to keep the DNA contour length constant within =2%. The number of monomers units, M, the bead diameter size b0, and the bond spinning diffusion coefficient are determined for each choice of the DNA length and radius as illustrated in the Results and Discussion section. The value of the torsional rigidity C (appearing in the BD as the parameter) and the value of the persistence length P (defined through the parameter) are assigned to each angle.

Results and Discussion A. Fragment Characterization. Circular dichroism data of the reference F0 and of the bulged fragment F2 are shown in Figure 1 from 320 to 200 nm. For both samples, a B-structure pattern is apparent though a small shift is found in the positive to negative crossing points of the ellipticity, being at ≈260 nm for F0 and at ≈264 nm for F2. This shift corresponds to a lower ellipticity of the fragment F2 that has five unpaired bases.29 A notable difference between the two fragments is found when the thermal stability of the double helix is measured. Under the presence of 5 mM magnesium added to the buffer, both fragments are found to keep their bases paired up to fairly high temperatures (T > 70 °C), with F0 showing a “melting” temperature of 82.7 ( 0.3 °C, while F2 melts at 75.6 ( 0.15 °C, as can be seen in Figure 2. Calf thymus DNA (nearly the same CG content as F0, 64%) in the same buffer melts at 85 ( 0.3 °C. From the values of the enthalpy and entropy changes derived from the melting profile of each fragment, one estimates that the enthalpy difference for the two fragments is ∆HF0 - ∆HF2 ) 11.3 ( 1 kcal/mol. Since the disruption of a base pair requires 1.6-1.8 kcal/ mol,29,30 then the lower value of the enthalpy change for F2 is compatible with fewer paired bases, six to seven, when compared to F0. Since F2 has 55 base pairs, while F0 has 50, it is likely that the “bubble” in the middle of the fragment can start nucleation of the denaturation and that probably 2 or 3 base pairs adjacent to the bubble are not paired. In particular when the dependence of the melting temperature on fragment length31 is being estimated as 1/TM ) a + b/n, where n is the number of base pairs, the melting temperature difference between a 50- and a 25-base pair

Figure 3. FPA data for the reference F0 (circles), the singleinserted F1 (triangles), and the double-inserted F2 (squares) fragments. Phase differences, lower data, and modulation ratios, upper data, are shown versus the modulation frequency. The continuous lines represent Brownian dynamics simulations fitted to the experimental data, according to the following parameters: for F0, R ) 1.1 nm, P ) 200 nm, C ) 2.55 × 10-19 erg, r0 ) 0.367, and IF ) 0.02; for F1, R ) 1.1 nm, P ) 200 nm, C ) 2.55 × 10-19 erg, r0 ) 0.33, IF ) 0.02, and bending angle θ ) 45°; for F2, R ) 1.1 nm, P ) 200 nm, C ) 2.55 × 10-19 erg, PB ) 1000 nm, CB ) 0.64 × 10-19 erg, r0 ) 0.37, and IF ) 0.017.

duplex, ≈12 °C, is expected to be compared to the observed difference, ≈7 °C. The bulged fragment considered equivalent to two 25-bp independent arms is certainly an oversimplification, but it helps in understanding how a nucleation in the middle can affect the thermal stability of the sample.10 The linking of the two arms is expected to increase the melting temperature, and this is in agreement with our experimental findings. B. Fluorescence Polarization Anisotropy. FPA data in the frequency domain are shown in Figure 3. Both phase difference and demodulation ratios are found to be higher for fragment F2 than for the reference fragment F0. Also shown in the figure for comparison are the FPA data for the single-bulged fragment, labeled F1, previously studied in our laboratory. This fragment, which has the extra five-adenine run on one strand only, was shown3 to be bent by ≈45°.

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The curious feature of the FPA data is that the two modified fragments have opposite trends with respect to the reference. As already recognized for the bent fragment, it will be shown that this behavior is related to both the dynamic and structural features of the samples. In this way, an inspection of the phase and modulation data allows a preliminary qualitative prediction of the dynamical properties of the samples. The continuous lines in Figure 3 represent best fits obtained by BD simulations and will be discussed later. The analytical fit of the FPA decay by means of the Schurr model (eq 1) gives information on fragment properties such as radius and torsional rigidity constant R ) C/h (h is the base pair thickness) and on the intercalated dye features, i.e., the limiting anisotropy r0 and the free dye fractional intensity IF.1,2,13 For the reference fragment F0, a radius R ) 1.08 ( 0.03 nm and a torsional constant R ) (7 ( 1) × 10-12 erg result. This kind of analysis had been previously performed also on the kinked fragment F1, when properly taking into account the rigid body spinning and tumbling diffusion coefficients, modified according to the fragment bending. As a result, an estimate of the bending angle, θ ≈ 45°, was obtained.3 The same fitting, applied to the double-bulged fragment F2, yields radius values close (although slightly smaller) to that of the reference and lower values for the torsional constant R ) (6 ( 1) × 10-12 erg . It is therefore likely that a change of the rigid body rotations is not the most relevant feature and that the internal dynamics of the F2 fragment is different from that of the reference fragment. This shows that FPA measurements are sensitive to different types of anomalies in short DNA fragments. We try in the following to rationalize the possible effect of the “bubble” on the DNA internal dynamics by analyzing the FPA data. This is somehow intriguing since the opposite behavior of the two “anomalous” fragments F1 and F2 could indicate the distinction between a static and a dynamic character of their curvature in solution. Before attempting a description of the data by means of time-consuming and sophisticated Brownian dynamics simulations, one could attempt to analyze the fluorescence anisotropies with a simple and essential model. C. Flexible Joint Model. We have seen above that a preliminary analysis of the FPA data suggests a perturbation of the internal dynamics of the F2 fragment with respect to F0. Due to the F2 base sequence, one expects a double bulge in the middle of two 25-bp arms, a structure that can be modeled by two cylindrical arms linked by a softer joint. The flexibility of this joint has two components: torsion, which allows one rod to spin around its axis, and bending, which changes the angle between the two rods. In order to give a simple analytical expression of the FPA decay according to this tentative model, it has been assumed that the “rigid” body decay is that corresponding to the equilibrium conformation of the fragment. Permanent bents induce a change mostly in the spinning diffusion coefficient, while dynamic anomalies such as a soft joint are mimicked by damped bending and torsional springs which simply add to the other depolarization contributions. A highly flexible joint should make the coupling of the internal dynamics of the two halves of the DNA fragment very weak: we assume therefore, as a crude approximation, that the internal dynamics of DNA can be described by that of the two separated halves.

Flexibility of a Bulged DNA Fragment 699

In general, the FPA decay can be written as a sum of torsional and bending depolarization factors6 of the type Cn(t) ) e-n2〈∆φT(t)2〉 and Fn(t) ) e-(6-n2)〈∆φB(t)2〉 (see eq 1), where ∆φT(t) and ∆φB(t) are the angular displacements experienced by the dye, due to the torsional and bending dynamics of the DNA, respectively. Each of these displacements is composed of a contribution from the internal DNA dynamics,2,5 from the rigid rod diffusion and, likely, from the flexible joint dynamics:

〈∆φT(t)2〉 ) DSPINt + 〈∆φinternal (t)2〉 + 〈∆φJOINT (t)2〉 T T (t)2〉 + 〈∆φJOINT (t)2〉 〈∆φB(t)2〉 ) DTUMBt + 〈∆φinternal B B (10) All together this leads to polarization losses of the fluorescence light of the type (see eq 1): 2

r(t) ) r0

∑ e-n D n)0 2

CJ,n(t)Cn(t)e-(6-n )DTUMBtFJ,n(t)Fn(t)

SPINt

2

(11)

where the functions Cn(t) and Fn(t) are the torsional and bending anisotropy contributions, respectively, due to the internal dynamics of each half of the molecule. In particular, CJ,n(t) and FJ,n(t) are the contributions from the joint,6 CJ,n(t) ) e-n2〈∆φT(t)2〉 and FJ,n(t) ) e-(6-n2)〈∆φB(t)2〉. Their mean square displacements can be written as32

〈∆φB(t)2〉 ) 2σB2(1 - e-tΓB)

(12a)

〈∆φT(t)2〉 ) 2σT2(1 - e-tΓT)

(12b)

where the two relaxation rates ΓB and ΓT are closely related to the variance of the angular displacements σB and σT and to spinning and tumbling diffusion coefficients D0SPIN and D0TUMB of the two DNA halves as ΓT ) D0SPIN/σT2 and ΓB ) D0TUMB/σB2. The parameters affecting the depolarization anisotropy are therefore the DNA torsional constant R, its length and its radius. DSPIN amd DTUMB are the overall spinning and tumbling diffusion coefficients, respectively: they are either computed according to the Garcia de la Torre equations for a rigid rod27 or used as free parameters of the fit. Two additional parameters are the bending and torsional variances of the angular displacements of the dye due to the motion about the joint. In fitting the FPA data from the “straight” fragment F0 used as reference, we used eq 11 with no joint, i.e., CJ,n(t) ) 1, FJ,n(t) ) 1, and (N + 1) ) 50. The FPA data from the F2 and F1 fragments are then fitted by keeping the same radius R, and same torsional constant as found for the straight fragment, but by assuming the presence of a joint and/or by fitting the tumbling and spinning diffusion coefficients. The results are summarized in Table 1. When the bulged F2 fragment is analyzed, the parameters found for the flexible joint are given by σT2 ) 0.014, which corresponds to an equivalent torsional rigidity of C ) hR ) h (KBT)/σT2 = 3 × 10-20 erg cm, almost 7-fold smaller than the usual one, while the low value for the bending angle variance σB2 ≈ 10-3 indicates that with this simple and approximated model no appreciable bending of the joint is apparent. For comparison, a fit of the F1 fragment data yields a

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Table 1. Fragment Parameters for FPA Fits by the Flexible Joint Model: DNA Radius R, Torsional Elastic Constant r, Spinning DSPIN and Tumbling DTUMB Diffusion Coefficients, Torsional σT and Bending σB Angular Displacement Variances due to the Flexible Joint, and Reduced χ2 a fragment

R (nm)

R × 1012 (erg)

DSPIN (MHz)

DTUMB (MHz)

σT2

σB2

χ2

F0 (straight) F2 (bulged) F1 (bent)

1.1 1.1* 1.1*

7.8 7.8* 7.8*

17.2 17.2 11.8

1.35 1.35 1.61

0.023